gaokao

Papers (66)
2025
national-I 19 national-II 18 national-I_eol 19
2024
beijing 21 national-II 19 national-I_github 19
2023
national-A-science 18 national-B-science 16 national-II 8
2022
national-A-arts 15 national-A-science 14 national-B-arts 20 national-B-science 17 national-I 19 national-II 7
2021
national-A-arts 19 national-A-science 18 national-I 14 national-II 12
2020
national-I-arts 20 national-I-science 9 national-II-science 17 national-III-arts 22 national-III-science 16 shanghai 17
2019
beijing-science 15 national-I-arts 13 national-I-science 12 national-I-science_gkztc 14 national-II-arts 13 national-II-science 9 national-II-science_gkztc 13 national-III-arts 14 national-III-science 16 national-III-science_gkztc 14
2018
national-I-arts 14 national-I-science 13 national-II-arts 20 national-II-science 18 national-III-science 17
2017
national-I-arts 13 national-I-science 14 national-II-arts 15 national-II-science 8
2016
national-I 7
2015
anhui-arts 14 beijing-arts 15 beijing-science 13 chongqing-arts 20 chongqing-science 15 fujian-science 18 hubei-arts 16 hubei-science 15 hunan-arts 21 hunan-science 11 jiangsu 19 national-II-arts 15 national-II-science 16 shaanxi-science 17 sichuan-arts 15 sichuan-science 21 tianjin-arts 17 tianjin-science 17 zhejiang-arts 15 zhejiang-science 15
Unknown
sample_questions 7
2015 tianjin-science

17 maths questions

Q1 5 marks Probability Definitions Set Operations View
Given the universal set $U = \{1,2,3,4,5,6,7,8\}$, set $A = \{2,3,5,6\}$, set $\mathrm{B} = \{1,3,4,6,7\}$, then $\mathrm{A} \cap \mathrm{C}_{U}\mathrm{B} =$
(A) $\{2,5\}$
(B) $\{3,6\}$
(C) $\{2,5,6\}$
(D) $\{2,3,5,6,8\}$
Q2 5 marks Inequalities Linear Programming (Optimize Objective over Linear Constraints) View
Variables $x, y$ satisfy the constraints $\left\{\begin{array}{c}x + 2 \geq 0, \\ x - y + 3 \geq 0, \\ 2x + y - 3 \leq 0,\end{array}\right.$ then the maximum value of the objective function $Z = x + 6y$ is
(A) 3
(B) 4
(C) 18
(D) 40
Q4 5 marks Inequalities Sufficient/Necessary Conditions Between Inequality Conditions View
Let $x \in \mathbb{R}$. Then ``$|x - 2| < 1$'' is ``$x^2 + x - 2 > 0$'' a
(A) sufficient but not necessary condition
(B) necessary but not sufficient condition
(C) necessary and sufficient condition
(D) neither sufficient nor necessary condition
Q5 5 marks Circles Chord Length and Chord Properties View
As shown in the figure, in circle O, M and N are trisection points of chord AB. Chords CD and CE pass through points M and N respectively. If $\mathrm{CM} = 2$, $\mathrm{MD} = 4$, $\mathrm{CN} = 3$, then the length of segment NE is
(A) $\frac{8}{3}$
(B) 3
(C) $\frac{10}{3}$
(D) $\frac{5}{2}$
Q6 5 marks Conic sections Equation Determination from Geometric Conditions View
Given the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ $(a > 0, b > 0)$, one of its asymptotes passes through the point $(2, \sqrt{3})$, and one focus of the hyperbola lies on the directrix of the parabola $y^2 = 4\sqrt{7}x$. Then the equation of the hyperbola is
(A) $\frac{x^2}{21} - \frac{y^2}{28} = 1$
(B) $\frac{x^2}{28} - \frac{y^2}{21} = 1$
(C) $\frac{x^2}{3} - \frac{y^2}{4} = 1$
(D) $\frac{x^2}{4} - \frac{y^2}{3} = 1$
Q7 5 marks Exponential Functions Ordering and Comparing Exponential Values View
Given the function $\mathrm{f}(x) = 2^{|x-1|} - 1$ defined on $\mathbb{R}$ (where m is a real number) is an even function, let $\mathrm{a} = \mathrm{f}(\log_{0.5}3)$, $b = f(\log_2 5)$, $c = f(2m)$. Then the size relationship of $a, b, c$ is
(A) $a < b < c$
(B) $a < c < b$
(C) $c < a < b$
(D) $c < b < a$
Q8 5 marks Curve Sketching Number of Solutions / Roots via Curve Analysis View
Given the function $F(x) = \left\{\begin{array}{l}2 - |x|, \quad x \leq 2 \\ (x - 2)^2, \quad x > 2\end{array}\right.$ and function $g(x) = b - f(2 - x)$, where $b \in \mathbb{R}$. If the function $y = f(x) - g(x)$ has exactly 4 zeros, then the range of $b$ is
(A) $\left(\frac{7}{4}, +\infty\right)$
(B) $\left(-\infty, \frac{7}{4}\right)$
(C) $\left(0, \frac{7}{4}\right)$
(D) $\left(\frac{7}{4}, 2\right)$
Q9 5 marks Complex Numbers Arithmetic Systems of Equations via Real and Imaginary Part Matching View
i is the imaginary unit. If the complex number $(1 - 2i)(a + i)$ is a pure imaginary number, then the real number a equals \hspace{2cm}.
Q11 5 marks Areas by integration View
The area of the closed figure enclosed by the curve $\mathrm{y} = \mathrm{x}^2$ and the line $\mathrm{y} = \mathrm{x}$ is \hspace{2cm}.
Q12 5 marks Binomial Theorem (positive integer n) Find a Specific Coefficient in a Single Binomial Expansion View
In the expansion of $\left(x - \frac{1}{4x}\right)^6$, the coefficient of $x^2$ is \hspace{2cm}.
Q13 5 marks Sine and Cosine Rules Find a side length using the cosine rule View
In $\triangle \mathrm{ABC}$, the angles $\mathrm{A}, \mathrm{B}, \mathrm{C}$ and their opposite sides $\mathrm{a}, \mathrm{b}, \mathrm{c}$ respectively. Given that the area of $\triangle \mathrm{ABC}$ is $3\sqrt{15}$, $b - c = 2$, $\cos A = -\frac{1}{4}$, then the value of $a$ is \hspace{2cm}.
Q14 5 marks Vectors Introduction & 2D Optimization of a Vector Expression View
In isosceles trapezoid ABCD, $\mathrm{AB} \parallel \mathrm{DC}$, $\mathrm{AB} = 2$, $\mathrm{BC} = 1$, $\angle \mathrm{ABC} = 60°$. Moving points E and F are on segments BC and DC respectively, with $\overrightarrow{\mathrm{BE}} = \lambda\overrightarrow{\mathrm{BC}}$, $\overrightarrow{\mathrm{DF}} = \frac{1}{9\lambda}\overrightarrow{\mathrm{DC}}$. Then the minimum value of $\overrightarrow{\mathrm{AE}} \cdot \overrightarrow{\mathrm{AF}}$ is \hspace{2cm}.
Q15 13 marks Trig Graphs & Exact Values View
Given the function $f(x) = \sin^2 x - \sin^2\left(x - \frac{\pi}{6}\right)$, $x \in \mathbb{R}$.
(I) Find the minimum positive period of $f(x)$;
(II) Find the maximum and minimum values of $f(x)$ on the interval $\left[-\frac{\pi}{3}, \frac{\pi}{4}\right]$.
Q16 13 marks Discrete Probability Distributions Deriving or Identifying a Probability Distribution from a Random Process View
To promote the development of table tennis, a certain table tennis competition allows athletes from different associations to form teams. There are 3 athletes from Association A, of which 2 are seeded players, and 5 athletes from Association B, of which 3 are seeded players. Randomly select 4 people from these 8 athletes to participate in the competition.
(I) Let A be the event ``exactly 2 seeded players are selected, and these 2 seeded players are from the same association''. Find the probability of this event.
(II) Let X be the number of seeded players among the 4 selected people. Find the probability distribution and mathematical expectation of the random variable X.
Q17 13 marks Vectors 3D & Lines Multi-Part 3D Geometry Problem View
As shown in the figure, in the quadrangular prism $\mathrm{ABCD} - A_1B_1C_1D_1$, the lateral edge $AA_1 \perp$ base $\mathrm{ABCD}$, $\mathrm{AB} \perp \mathrm{AC}$, $\mathrm{AB} = 1$, $\mathrm{AC} = AA_1 = 2$, $AD = CD = \sqrt{5}$, and points M and N are the midpoints of $B_1C$ and $D_1D$ respectively.
(I) Prove: $\mathrm{MN} \parallel$ plane ABCD
(II) Find the sine value of the dihedral angle $D_1 - AC - B_1$;
(III) Let E be a point on edge $A_1B_1$. If the sine value of the angle between line NE and plane ABCD is $\frac{1}{3}$, find the length of segment $A_1E$.
Q18 13 marks Sequences and series, recurrence and convergence Closed-form expression derivation View
Given the sequence $\{a_n\}$ satisfies $a_{n+2} = qa_n$ (where q is a real number and $q \neq 1$), $n \in \mathbb{N}^*$, $a_1 = 1$, $a_2 = 2$, and $a_2 + a_3$, $a_3 + a_4$, $a_4 + a_5$ form an arithmetic sequence.
(I) Find the value of q and the general term formula of $\{a_n\}$;
(II) Let $b_n = \frac{\log_2 a_{2n}}{a_{2n-1}}$, $n \in \mathbb{N}^*$. Find the sum of the first n terms of the sequence $\{b_n\}$.
Q19 14 marks Circles Chord Length and Chord Properties View
Given an ellipse with left focus $\mathrm{F}(-c, 0)$ and eccentricity $\frac{\sqrt{3}}{3}$. Point M is on the ellipse and in the first quadrant. The line segment of line FM intercepted by the circle $x^2 + y^2 = \frac{b^2}{4}$ has length c, and $|FM| = \frac{4\sqrt{3}}{3}$.
(I) Find the slope of line FM;
(II) Find the equation of the ellipse;
(III) Let P be a moving point on the ellipse. If the slope of line FP is greater than $\sqrt{2}$, find the range of the slope of line OP